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KANSAS STATE UNIVERSITY Manhattan, Kansas CONSTRAINING COMPETING MODELS OF DARK ENERGY WITH COSMOLOGICAL OBSERVATIONS by ANATOLY PAVLOV B.S., Saint Petersburg State Polytechnic University, Russia 2005 M.S., Saint Petersburg State Polytechnic University, Russia 2007 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2015 Abstract The last decade of the 20th century was marked by the discovery of the accelerated ex- pansion of the universe. This discovery puzzles physicists and has yet to be fully understood. It contradicts the conventional theory of gravity, i.e. Einstein's General Relativity (GR). According to GR, a universe filled with dark matter and ordinary matter, i.e. baryons, leptons, and photons, can only expand with deceleration. Two approaches have been developed to study this phenomenon. One attempt is to assume that GR might not be the correct description of gravity, hence a modified theory of gravity has to be developed to account for the observed acceleration of the universe's expansion. This approach is known as the "Modified Gravity Theory". The other way is to assume that the energy budget of the universe has one more component which causes expansion of space with acceleration on large scales. Dark Energy (DE) was introduced as a hypothetical type of energy homogeneously filling the entire universe and very weakly or not at all interacting with ordinary and dark matter. Observational data suggest that if DE is assumed then its contribution to the energy budget of the universe at the current epoch should be about 70% of the total energy density of the universe. In the standard cosmological model a DE term is introduced into the Einstein GR equations through the cosmological constant, a constant in time and space, and proportional to the metric tensor gµν. While this model so far fits most available observational data, it has some significant conceptual shortcomings. Hence there are a number of alternative cosmological models of DE in which the dark energy density is allowed to vary in time and space. CONSTRAINING COMPETING MODELS OF DARK ENERGY WITH COSMOLOGICAL OBSERVATIONS by ANATOLY PAVLOV B.S., Saint Petersburg State Polytechnic University, Russia 2005 M.S., Saint Petersburg State Polytechnic University, Russia 2007 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2015 Approved by: Major Professor Bharat Ratra Copyright ANATOLY PAVLOV 2015 Abstract The last decade of the 20th century was marked by the discovery of the accelerated ex- pansion of the universe. This discovery puzzles physicists and has yet to be fully understood. It contradicts the conventional theory of gravity, i.e. Einstein's General Relativity (GR). According to GR, a universe filled with dark matter and ordinary matter, i.e. baryons, leptons, and photons, can only expand with deceleration. Two approaches have been developed to study this phenomenon. One attempt is to assume that GR might not be the correct description of gravity, hence a modified theory of gravity has to be developed to account for the observed acceleration of the universe's expansion. This approach is known as the "Modified Gravity Theory". The other way is to assume that the energy budget of the universe has one more component which causes expansion of space with acceleration on large scales. Dark Energy (DE) was introduced as a hypothetical type of energy homogeneously filling the entire universe and very weakly or not at all interacting with ordinary and dark matter. Observational data suggest that if DE is assumed then its contribution to the energy budget of the universe at the current epoch should be about 70% of the total energy density of the universe. In the standard cosmological model a DE term is introduced into the Einstein GR equations through the cosmological constant, a constant in time and space, and proportional to the metric tensor gµν. While this model so far fits most available observational data, it has some significant conceptual shortcomings. Hence there are a number of alternative cosmological models of DE in which the dark energy density is allowed to vary in time and space. Table of Contents List of Figures ix List of Tables xiv Acknowledgements xiv 1 Introduction1 1.1 Theoretical Cosmology . .1 1.1.1 Short overview of General Relativity . .1 1.1.2 Friedmann Equations . .4 1.1.3 Solutions of Friedmann Equations . .8 1.1.4 Cosmological Constant . 13 1.1.5 Standard Cosmological Model . 16 1.2 Observational Cosmology . 17 1.2.1 Redshift . 17 1.2.2 Hubble parameter . 18 1.2.3 Luminosity Distance . 19 2 Forecasting cosmological parameter constraints from near-future space- based galaxy surveys 23 2.1 Introduction . 23 2.2 Measured power spectrum of galaxies . 28 2.3 Cosmological models . 32 vi 2.3.1 ΛCDM, XCDM and !CDM parameterizations . 32 2.3.2 φCDM model . 33 2.4 Fisher matrix formalism . 34 2.5 Results . 36 2.5.1 Constraints on growth rate . 36 2.5.2 Constraints on dark energy model parameters . 37 3 Nonflat time-variable dark energy cosmology 47 3.1 Introduction . 47 3.2 The Model . 49 3.2.1 Solution for the curvature-dominated epoch . 51 3.3 Some observational predictions . 54 3.3.1 The time parameter H0t0 ......................... 55 3.3.2 The distance modulus difference ∆m(z)................ 57 3.3.3 Number counts . 59 3.3.4 The growth of structure . 59 4 Cosmological constraints from large-scale structure growth rate measure- ments 63 4.1 Introduction . 63 4.2 Data and analysis . 65 4.2.1 Growth rate of LSS . 65 4.2.2 SNIa distance modulus . 68 4.2.3 Hubble parameter . 68 4.2.4 Computation of joint χ2(p)....................... 69 4.3 Results and discussion . 70 vii 5 Conclusion 74 Bibliography 85 A Appendix for Chapter 2 86 B Appendix for Chapter 3 90 viii List of Figures 1.1 Light cone diagram. .3 1.2 Cosmic Microwave Background radiation map of the last scattering epoch. Data of 2012 WMAP mission, Source: http://map.gsfc.nasa.gov/media/060913/index.html5 1.3 Two dimensional analogies of hyperbolic, flat, and spherical manifolds. .7 1.4 Solutions of Friedmann equations for a single component universes with flat, closed, and open geometries. The time and the scale factor in arbitrary units. 12 1.5 Numerical solution of Eq. (1.30) for standard cosmology. Time and the scale factor presented in arbitrary units, where a = 1 corresponds to the current epoch. ....................................... 15 1.6 The plot of log10(ρi/ρcr) vs the log of the scale factor log10(a) for each compo- nent. One can see that the early universe was dominated by radiation, how- ever since energy density of radiation dissipates as a−4 while energy density of ordinary matter decreases as a−3, the radiation dominated epoch switches to matter dominated. Eventually, the universe becomes dominated by the cosmological constant, since its energy density does not evolve in time. 16 1.7 Hubble diagram for the Union2.1 compilation. The solid line represents the best-fit cosmology for a flat ΛCDM universe for supernovae alone. Source: Fig.(4) of [12]................................... 22 ix 2.1 Predicted relative error on the measurements of growth rate as a function of redshift z in redshift bins of ∆z = 0:1 for different models of dark energy. The upper solid black line shows predictions for the case when no assumption is made about the nature of dark energy. 40 2.2 Predicted one standard deviation confidence level contour constraints on the current renormalized Hubble constant h and the parameter γ that describes deviations from general relativity for different dark energy models. 41 2.3 Upper panel shows one standard deviation confidence level contours con- straints on parameters !a and !0 of the !CDM parametrization, while lower panel shows these for parameters Ωk and Ωm.................. 42 2.4 One standard deviation confidence level contour constraints on parameters of the XCDM parametrization. 43 2.5 One standard deviation confidence level contour constraints on parameters α and Ωm of the φCDM model. Lower panel shows a magnification of the tightest two contours in the lower left corner of the upper panel. 44 2.6 One standard deviation confidence level contour constraints on parameters Ωk and Ωm of the φCDM model. The lower panel shows a magnification of the two tightest contours in the center of the upper panel. 45 2.7 One standard deviation confidence level contour constraints on parameters Ωk and Ωm of the ΛCDM model. 46 x 3.1 Contours of fixed time parameter H0t0, as a function of the present value of the nonrelativistic matter density parameter Ωm0 and scalar field poten- tial power-law index α, at various values of the current value of the space- curvature density parameter Ωk0 (as listed in the inset legend boxes). The upper panel shows a larger part of (Ωm0, α) space for a larger range of Ωk0 values [for H0t0 = 0:7; 0:75; 0:8; 0:85; 0:95; 1:05 and 1:15, from right to left], while the lower panel focuses on a smaller range of the three parameters [for H0t0 from 0:8 to 1:15 in steps of 0:05, from right to left]. 56 3.2 Contours of fixed bolometric distance modulus relative to the Einstein{de Sitter model, ∆m(z = 1:5), as a function of the matter density parameter Ωm0 and scalar field potential power-law index α, and various values of the space curvature density parameter Ωk0 (as listed in the inset legend boxes).
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